To initiate the “top-down, least-change” localization procedure (see Sec. REF ), the molecule is partitioned into four fragments (see Fig. REF ), with the dangling bonds saturated with the modified[1]} PHO (projected hybrid orbital) boundary atoms[2]}. While subsystems 1-3 are obviously closed-shell systems,
subsystem 4 can be in singlet, triplet or quintet. After the HF/ROHF calculations of
these subsystems, the orthonormal pFLMOs are generated[3]}, [4]}, which are taken as the initial guess for the HF/ROHF calculations of the whole systems.
The FLMOs are finally generated according to the “top-down, least-change” scheme. Since such FLMOs
have one-to-one correspondence with the pFLMOs, the active orbitals can be chosen simply from the subsystem pFLMOs. However, to make the \(\mathbb {i}\) CAS calculations
“more difficult”, we still start with valence AOs. Looking at the chemical structure of the system, it should be easy to guess that, apart from
the five 3d orbitals of Co, the five 2p\(_z\)  orbitals of the C and N atoms of the 5-membered ring as well as the 2p\(_x\)  and 2p\(_z\)  of N\(_{\textrm {imido}}\) 
are most relevant for the chemical environment of Co, since the 2p\(_y\)  orbital of N\(_{\textrm {imido}}\)  participates in the bonding with the 6-membered ring.
In addition to these valence AOs, the five 4d orbitals of Co can further be included to account for the so-called “double d-shell effect”
resulting from physical intuitions[5]}.
In essence, putting Co 4d into the active space amounts to shifting the inaccurate, perturbative treatment of their
radial correlation effect to the exact, variational treatment. We then have in total 17 valence AOs (and hence 17 pre-LMOs/CMOs) along with 16 active electrons
(6 for Co\(^{3+}\)  (3d\(^6\) ), 6 for the anionic 5-membered ring, and 4 for \(:\) N\(_{\textrm {imido}}^{2-}\) )
according to minimal chemical and physical intuitions. Taking the 8 doubly occupied and 9 unoccupied pre-LMOs as probes, precisely
17 HF-LMOs (see Fig. REF ) can be selected as initial guess for \(\mathbb {i}\) CAS(16,17).
As can be seen from Tables REF  and REF ,
the large absolute overlaps between the pre-LMOs and HF-LMOs do support the selection. The same findings also hold for the triplet and quintet states (see
Supporting Information). Nevertheless, the CAS(16,17) calculations are too expensive (which, however, can readily be
performed with the iCISCF algorithm[6]} that employs the selected iterative configuration interaction (iCI)[7]}, [8]}, [9]} as the CAS solver).
The first attempt to reduce CAS(16,17) is to exclude the five 2p\(_z\)  orbitals of the 5-membered ring, thereby leading to CAS(10,12).
As a matter of fact, this amounts just to deleting from Fig. REF  as well as Tables REF  and REF 
the \(\pi -1\) , \(\pi -2\) , \(\pi -2\) , \(\pi ^\ast -1\) , and \(\pi ^\ast -2\)  orbitals that are
localized on the 5-membered ring.
It was also attempted[10]} to further remove the 4d\(_{xy}\)  and 4d\(_{yz}\)  orbitals from CAS(10,12), for the 3d\(_{y^2}\) , 3d\(_{x^2-z^2}\)  and 3d\(_{xz}\) 
orbitals are doubly occupied in the singlet ground state, such that only 4d\(_{y^2}\) , 4d\(_{x^2-z^2}\)  and 4d\(_{xz}\)  are relevant for the “double d-shell effect”.
A more dramatic simplification is to exclude all the 4d orbitals from CAS(10,12), thereby leading to CAS(10,7).
These two cases amount to deleting from Fig. REF  the last two and five orbitals involving Co 4d, respectively.
